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**Read Online or Download Robin Hartshorne's Algebraic Geometry Solutions. Chapter II Section 7, Projective Morphisms PDF**

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Furthermore, L is a complete lattice if every subset of L has a supremum and an inﬁmum. We say that x is way below y and write x y if y ≤ ∨D for every directed set D implies x ≤ z for some z ∈ D. e. for each x ∈ L, the set {y ∈ L : y x} is directed with supremum x. A set U ⊂ L is called Scott open if U is an upper set, that is ↑ x = {y ∈ L : x ≤ y} ⊂ U for all x ∈ U , and if x ∈ U implies the existence of some y ∈ U with y x. The collection of all Scott open sets is a topology on L which is denoted by Scott(L).

It should be noted that a bounded random closed set X in a Polish space is not always Hausdorff approximable, even if the realisations of X are almost surely convex. 12 (Non-approximable random closed sets). Consider the probability space Ω = [0, 1] with the Lebesgue σ -algebra and the Lebesgue measure P. (i) Let E be the Banach space of real-valued continuous functions on [0, 1] with the uniform norm. Deﬁne a multifunction with closed convex values as X (ω) = {x ∈ E : x ≤ 1 , x(ω) = 0} for ω ∈ Ω.

Ii) Let E = 2 be the space of square-summable sequences. For each ω ∈ Ω = −n with ω equal 0 or 1. Let [0, 1] take its binary expansion ω = ∞ n n=1 ωn 2 X (ω) = {x ∈ 2 : x ≤ 1 , x n = 0 for ωn = 0, n ≥ 1} . Then X is not Hausdorff approximable, since ρH (X (ω), X (ω )) = 1 for ω = ω . 2 Selections of random closed sets Fundamental selection theorem Recall that S(X) denotes the family of all (measurable) selections of X. 3) implies the following existence theorem for selections. 13 (Fundamental selection theorem).